Higher order effects in electromagnetic dissociation of neutron halo nuclei

14 Pages·0.16 MB·English

by  S. Typel

Higher Order Effects in Electromagnetic Dissociation of Neutron
Halo Nuclei
S. Typel
National Superconducting Cyclotron Laboratory, Michigan State University,
East Lansing, Michigan 48824-1321, USA
G. Baur
1
Institut fu¨r Kernphysik, Forschungszentrum Ju¨lich, 52425 Ju¨lich, Germany
0
0 (February 8, 2008)
2

n
a
J
Abstract

6
1

1 We investigate higher order effects in electromagnetic excitation of neutron
v
halo nuclei using a simple and realistic zero range model for the neutron-
3
3 core interaction. In the sudden (or Glauber) approximation all orders in the
0 target-core electromagnetic interaction are taken into account. Small devia-
1
0 tions from the sudden approximation are readily calculated. We obtain very
1 simple analytical results for the next to leading order effects, which have a
0
simple physical interpretation. For intermediate energy electromagnetic dis-
/
h
sociation, higher order effects are generally small. We apply our model to
t
- Coulombdissocationof19Cat67AMeV.Theanalyticalresultsarecompared
l
c ·
to numerical results from the integration of the time dependent Schro¨dinger
u
n equation. Good agreement is obtained. We conclude that higher order elec-
:
v tromagnetic effects are well under control.
i
X
25.70.De, 25.70.Mn, 25.60.-t
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Typeset using REVTEX
1
I. INTRODUCTION
Electromagnetic excitation of high energy radioactive beams is a powerful method to
study electromagnetic properties of loosely bound neutron rich nuclei. E.g., the low lying
E1-strengths of one-neutron halo nuclei like 11Be and 19C have been studied in this way
[1–3]. In a similar way, two-neutron halo nuclei like 6He and 11Li were studied. Such
experiments are usually analysed theoretically in first order electromagnetic perturbation
theory or the equivalent photon method. In this way, the multipole (especially dipole)
strength distribution is obtained. Such an analysis depends on the dominance of first order
excitations. Various methods have been developed in order to consider deviations from first
order perturbation theory with the usual multipole expansion of the interaction. However,
a consistent picture of the importance of these approximations has still not emerged.
By“higherordereffects” wemeanonlyelectromagnetically inducedeffectsontherelative
momentum of the fragments. They can be studied in the semiclassical approximation. In
the widest sense all effects which give rise to a deviation from the result of the traditional
semiclassical first order perturbative calculation of Coulomb breakup can be summarized
under this expression. In the perturbative approach higher order effects can be described
as the exchange of more than one photon between the target and the projectile system.
“Postacceleration” is also a higher order effect. In a classical picture it can be understood
as a different acceleration of the fragments in the Coulomb field of the target which will
change both the c.m. momentum and the relative momentum of the particles in the final
state. In our calculations we will not treat quantal effects like diffraction or contributions
to the breakup from the nuclear interaction.
There are mainly two different approaches for the investigation of higher order effects. In
the semiclassical description the projecile moves on a classical trajectory (which is usually
well justified) and experiences a time-dependent interaction from the target. Only the
excitation of the projectile is treated quantally. In contrast to that, the total system of
target, projectile and fragments, respectively, can be described by suitable wave functions in
a fully quantum mechanical approach. Each of these approaches has its merits, but, at the
same time, can limit the study of certain higher order effects or make it difficult to extract
them by a comparison to a suitable first order calculation.
The breakup of the prototype of a loosely bound nucleus, the deuteron, has for a long
time been studied in the post-form DWBA theory. Later, it has also been applied to neutron
halo (core + neutron) nuclei like 11Be. In this approach, the Coulomb interaction between
the target and the core is taken into account to all orders. This is done by using full
Coulomb wave functions in the initial and final state. For a recent review with further
references see [4]. A so called adiabatic breakup theory has recently been developed in [5].
This model is related to the post-form DWBA. It leads to a very similar formula, however,
the physical interpretation is somewhat different. Without entering into the differences of
the two approaches, it is clear that in these theories higher order effects are automatically
included to all orders. It is therefore very interesting to note that Tostevin [6] claims to have
found substantial higher order effects in the Coulomb breakup of 19C [3]. He compared his
results from the adiabatic approach to the one using semiclassical first order theory. Both
calculations have very similar relative energy spectra, but they differ by about 35 percent
in absolute magnitude.
2
It is the purpose of this paper to investigate higher order effects in the electromagnetic
excitation of neutron halo nuclei by comparing lowest order and higher order approxima-
tions within the same model. This is expected to give more reliable statements about the
importance of these effects than the comparison of higher order calculations in one theory
with first order calculations in another theory, where, e.g., the finite range effects are treated
in another way or the semiclassical approximation is not applied. We will limit ourselves
to the semiclassical description considering only the Coulomb interaction and will not in-
vestigate nuclear induced effects. In our approach we use a classical trajectory to describe
the relative motion between the target and the projectile. It should be kept in mind that
there is some ambiguity in the definition of this trajectory. The energy loss should be small
compared to the total kinetic energy of the projectile and some averaging procedure can be
used. We assume that the c.m. of the projectile moves on the classical trajectory (straight
line or Rutherford). It has been argued that the electromagnetic interaction of the target
only affects the charged core of the projectile; therefore the c.m. of the core has been used
for describing the classical motion. However, for intermediate energy this effect was found
to be quite small in numerical calculations in [7]. Actually, the result of a first order E1
calculation does not depend on this choice, since the dipole moment of the system does not
change. There is only a change of the quadrupole moment but this has small effects since
the E2 contribution to the breakup is rather small (see below). Since the c.m. trajectory
is fixed, only higher order effects in the relative motion of the fragments can be handled in
the semiclassical approach. Since the total momentum of the fragments is much larger than
the relative momentum between them, higher order effects have a much larger effect on the
relative momentum.
To some approximation the nuclear structure of neutron halo nuclei can be described
by rather simple wave functions. Using these wave functions the reaction mechanism can
be studied in a very transparent way and analytical results are obtained. In a later stage
more refined descriptions of the nuclei can be introduced. We recall some results from
[8] and apply the model to the electromagnetic breakup of 19C in comparison to more
accurate descriptions. In Ch. 2 the theoretical framework is given; results and comparison
to experimental results [3] are presented in Ch. 3. Conclusions are given in Ch. 4.
II. THEORETICAL FRAMEWORK
We follow very closely the approach of [8], see also [9]. In this straight-line semiclassical
model a projectile with charge +Ze impinges on a neutron+core (n+c) system with impact
parameter b and velocity v. The ground state wave function of the bound n+c system is
given by a simple Yukawa type wave function
η exp( ηr)
φ = − (2.1)
2π r
r
where the parameter η is related to the binding energy E by E = ¯h2η2 with the reduced
0 0 2m
mass m = mnmc of the system. The final continuum state is given by
mn+mc
1 exp( iqr)
φ(−) = exp(i~q ~r) − (2.2)
q · − η iq r
−
3
where the wave number q is related to the relative energy by E = ¯h2q2. With these wave
rel 2m
functionsthebreakupprobabilitycanbecalculatedanalyticallyinthesuddenapproximation
(corresponding to the Glauber or frozen nucleus approximation) of semiclassical Coulomb
excitation theory including all orders in the exchange of photons between the target and the
projectile. ButthetimeevolutionofthesytemduringtheexcitationisneglectedandonlyE1
transitions are taken into account. The first approximation corresponds to an adiabaticity
parameter ξ of zero. This quantity is the ratio of the collision time to the nuclear interaction
time and it is given by ξ = (E0+Erel)b. The multipole response of the system is characterized
¯hv
by effective charges Z(λ) = Z mn λ. They become very small for higher multipolarities
eff c mn+mc
due to the small ratio mn . (cid:16)From a(cid:17)perturbation expansion of the excitation amplitude it
mn+mc
can be shown that also the second order E1-E1-amplitude is much larger than the first order
E2 amplitude. The ratio is given by the Coulomb (or Sommerfeld) parameter ZZce2 which
¯hv
is much larger than one for high charge numbers Z. Therefore we can safely neglect E2
excitation in the following. (This is, e.g., qualitatively different for p+core systems like 8B
7Be + p with much larger E2 effective charges.) This can be considered as a justification
→
of the model of [8].
We expand the analytical results for the excitation probability of [8] for ξ = 0 up to
second order in E1 excitation or equivalently in the the characteristic strength parameter
which is given by
2ZZ(1)e2
χ = eff (2.3)
vbh¯k
where k = √η2 +q2. In leading order (LO), the sudden limit of the first order result of
[8] is obtained. Deviation for finite values of ξ can be calculated according to [8]. From
eq. (12) of [8] one sees that the ξ dependence of the amplitudes is given by ξK (ξ) or ξK (ξ)
1 0
with modified Bessel functions, respectively. For ξ = 0 this factor (for K ) is 1 and drops
1
to zero exponentially for ξ 1. The ξ-correction in the next-to-leading order (NLO) goes
≫
essentially like the square of this, so we can only overestimate the higher order effects in the
present procedure.
Instead of using the strength parameter χ we define in the following the slightly different
parameters
k
y = χ (2.4)
η
(independent of q) and
q E
x = = rel . (2.5)
η s E
0
AfterangularintegrationovertherelativemomentumbetweenthefragmentstheLObreakup
probability is found to be (see eq. 37 of [8])
dP 16 x4
LO = y2 . (2.6)
dq 3πη (1+x2)4
4
(Note that for a correct normalization of the breakup probability the results of [8] have to
be devided by (2π)3.) The NLO contribution is proportional to y4 and contains a piece from
the second order E1 amplitude and a piece from the interference of first and third order
amplitudes. Again, in terms of the variables η, x, and y one obtains
dP 16 x2(5 55x2 +28x4)
NLO = y4 − . (2.7)
dq 3πη 15(1+x2)6
The LO-expression is directly proportional to the B(E1)-strength with its characteristic
shape in the zero range model. The NLO-contribution will introduce a change of that
shape. It is weighted most in collisions with the smallest possible impact parameters b and
can easily be evaluated. For 0.309 < x < 1.367 the NLO contribution becomes negative with
the largest reduction at a relative energy close to the binding energy. This is essentially due
to the interference of first and third order amplitudes. The second order E1-E1 contribution
is positive definite. From [8] we find
dPE1−E1 = 16 y4x2(5+5x2 +16x4) . (2.8)
dq 3πη 15(1+x2)6
A reduction of the cross section at small relative energies is only obtained if third-order
contributions in the breakup amplitude are considered, either in a perturbative treatment
(cf. figs. 2 - 4 in [10]) or a full dynamical calculation (cf. figs. 5 + 7 in [11]). In our
analytical results we can directly see the dependence of higher order effects on the impact
parameterb, theprojectile velocity v andthebinding energy E charactrizedby η. Forlarger
0
impact parameters the first order E1 contribution will dominate more and more (y b−1).
∝
Perhaps, experimental accuracy will not be high enough to see such a change of the shape
of the breakup bump. The scaling variable y also displays very clearly the dependence
on the binding energy, characterized by η, and the charge number Z. Since Zc is about
mc
constant for all nuclei, the breakup probabilities for heavier nuclei (like the r-process nuclei)
are expected to be of the same order of magnitude as the light ones (like 11Be or 19C). This
will be an interesting field for future RIA facilities, where intensive beams of medium-energy
neutron-rich nuclei will become available. This is of special interest for the r-process [12].
The breakup cross section can be obtained by multiplying the differential breakup prob-
ability with the Rutherford scattering cross section dσR and the density of final states. It is
dΩ
given for the LO approximation by
d2σ dσ dP m
LO R LO
= (2.9)
dE dΩ dΩ dq h¯2q
rel
and similarly for the NLO approximation. In order to have a quick estimate of higher order
effects in the total breakup cross section we can integrate over the scattering angle and the
breakup relative energy
d2σ bmax dP
σ = dE dΩ = 2π dq dbb LO (2.10)
rel
dE dΩ dq
Z Z rel Z Z
bmin
where we have introduced minimum and maximum impact parameters b and b , re-
min max
spectively. We use b = ¯hv corresponding to a cutoff at an adiabaticity parameter of
max E0+Erel
5
ξ = 1. The integration over the impact parameter b is now easily performed. Introducing
the effective strength parameter
2ZZ(1)e2
χ = yηb = χkb = eff (2.11)
eff h¯v
and the minimal adiabaticity parameter
E b
ξ = 0 min (2.12)
min h¯v
we finally obtain
2
π χ
σ = eff [1 6ln(4ξ )] (2.13)
LO 18 η ! − min
and
4
π χ 23
σ = b−2 eff +18ξ2 , (2.14)
NLO −18 min η ! (cid:20)40 min(cid:21)
i.e., a reduction of the first order result. The total cross sections can only be a rough guide
because of the simple treatment of the cutoff. Modifications due to a more precise treatment
of the ξ-dependence have usually to be introduced (see below eq. (3.1)). However, the ratio
givesareasonableapproximationtothehigherordereffects. Itisasimplefunctiondepending
on the characteristic parameters of the excited system and the experimental conditions.
III. APPLICATION TO THE COULOMB BREAKUP OF 19C
In a recent experiment at RIKEN the breakup of 19C into 18C and a neutron scattered
on a Pb target with a beam energy of 67 A MeV was studied and the binding energy of the
·
neutron was determined to be 0.53 MeV [3]. We apply our model to this case since the high
beam energy together with the simple structure and small binding energy of the neutron is
favourable for a comparison. Another example would be 11Be where essentially the same
considerations apply.
Besides our simple analytical model we also present results from usual first order semi-
classical calculations and higher order calculation where the full time dependent Schr¨odinger
equation is solved for the 19C system which is perturbed by the time dependent Coulomb
field of the Pb nucleus. Here we have used the methods described in [13]. The neutron in
the bound state of 19C was assumed to be in a 2s state as deduced by Nakamura et al. [3].
1/2
The wave function was calculated assuming a Woods-Saxon potential of radius r = 3.3 fm
and diffuseness parameter a = 0.65 fm. The depth is adjusted to V = 39.77 MeV in order
−
to get the experimentally extracted binding energy of 0.53 MeV [3]. As in the analytical
model we use plane waves for the scattering wave functions in the final states for l > 0. The
wave functions are represented on a grid with exponential increasing mesh size similar to [7]
with 400 points up to a maximum radius of 900 fm.
In Fig. 1 we show the double differential cross section as a function of the relative energy
for three scattering angles. We have chosen 0.3◦, 0.9◦, and 2.7◦, which corresponds to
6
impact parameters of 109.7 fm, 36.6 fm, and 12.2 fm, respectively. In order to compare the
cross section in our model with finite-ξ results of the first order semiclassical calculation we
multiply the analytical cross section given in eq. (2.9) with the shape function
φ(ξ) = ξ2 K2(ξ)+K2(ξ) (3.1)
0 1
h i
of the photon spectrum and a normalization factor N. The function φ(ξ) gives the correct
dependence on the adiabaticity parameter in first order. We have φ(0) = 1 and it drops
to zero rapidly for ξ > 1. The factor N accounts for finite range effects. The ground
state wave function in the analytical model is a 1s1 state which has a different asymptotic
2
normalization but the same slope as compared to the corresponding wave function from the
Woods-Saxon potential. The results of the numerical calculation (dotted and dot-dashed
lines) agrees very well with the ξ-corrected cross section in the analytical model (solid and
dashed lines) for N = 2.73. The slope of the wave function is determined by the binding
energy (see eq. (2.1)) which is the same in both models. There are noticeable differences
between the analytical model and the dynamical calculation only for large relative energies
and scattering angles. The first order E2 contribution (multiplied with 1000) is also shown
in Fig. 1 (long-dashed line). It is at least three orders of magnitudes smaller as compared
to the first order E1 excitation cross section and can safely be neglected. Furthermore
we observe that the cross section decreases strongly with increasing scattering angle. For
small scattering angles results of first and higher order calculations are almost identical.
With increasing scattering angle we notice a reduction of the cross section for small relative
energies due to higher order electromagnetic effects.
InFig.2we comparetheratio ofhigher order (i.e., allorders inthedynamical calculation
or LO+NLO in the analyical model, respectively) to first order cross sections depending on
the relative energy for the same scattering angles as in Fig. 1. The solid line gives the
result of the analytical model. (Notice that the ratio is independent of φ(ξ) and N.) The
dependence of the ratio on the relative energy again agrees well with the ratio in the full
semiclassical model (dotted line). At small relative energies there is a reduction of the
cross section (except for energies close to zero) whereas at higher relative energies we find
a small increase. This behaviour can be directly understood by inspecting eqs. (2.6) and
(2.7). Higher order effects are largest for large scattering angles corresponding to impact
parameters close to grazing scattering. A look at the breakup probabilities (2.6) and (2.7)
shows that higher order effects increase essentially with b−2. The discrepancy of the two
models at higher relative energy, where the exact form of the wave function for small radii
in the range of the nuclear potential becomes important, is not very essential because the
absolute cross sections are very small. In contrast, at small relative energies, the models
agree very well since the main contribution to the matrixelements is determined by the
asymptotic form of the wave function.
◦ ◦
Integrating the double differential cross sections over scattering angles between 0 and 3
(b 11 fm) leads to the energy dependent cross sections in Fig. 3(a). Again we observe
min
≈
that both the first order and higher order calculations in the ξ-corrected analytical model
and the full semiclassical model agree very well in the peak of the distribution. Here we
find a reduction of the cross section of at most 10% at small relative energies as shown in
Fig. 3(b). The spectral shape is not severely distorted. Higher order Coulomb effects cannot
explain the difference of first order theoretical calculations and the experimental results with
7
respect to the absolute magnitude and the shape of the experimental data [3]. Although
they lead to a decrease of the cross section at small energies (apart from the region just
above threshold) and an increase at higher energies the slope of the theoretical results is
much steeper as compared to the experiment. In contrast, the position of the peak is well
described since it is determined by the binding energy of the neutron in 19C.
Integrating over relative energies the effects of higher order are washed out and become
even smaller in the total cross section σ. We obtain 1.44 b (1.39 b) in the first order (dynam-
ical) semiclassical calculation and 1.49 b (1.44 b) in the LO (NL+NLO) analytical model
with ξ-correction, respectively, for energies up to 3 MeV. Comparing to the experimental
value of σ = (1.34 0.12) b [3] one has to take into account our simple nuclear model. In
±
reality the ground state of 19C has a more complicated structure than a 2s single particle
1/2
state. Multiplying the cross sections of our calculation with a spectroscopic factor of 0.67 as
given by Nakamura et al. [3] the total cross section would be smaller than in the experiment
but the peak region in Fig. 3(a) would be well described. At higher relative energies nuclear
contributions could be present in the experimental data, increasing the total cross section
again. A possible Coulomb-nuclear interference effects could also lead to a change of shape
of the cross section. Furthermore, the experimental data could contain contributions from
final states with an excitation of the core 18C. Our results correspond to a reduction of the
total cross section by higher order effects of 3.3% in the semiclassical model and of 3.2%
in the ξ-corrected analytical model. From equations (2.13) and (2.14) we predict a 2.9%
reduction which is close to the more refined models. From the comparison we conclude that
our simple analytical model with finite-ξ correction is quite realistic in the prediction of
higher order effects and gives a reliable estimate of the reduction of the total cross section.
The smaller value of the reduction obtained in the simpler fully analytical model can be
well understood. Without taking the adiabatic suppression correctly into account contribu-
tions to the total cross section from higher relative energies and larger impact parameters,
where higher order effects are smaller, are not sufficiently reduced and lead to an underesti-
mate. However, higher order effects in the triple differential cross section in the peak of the
excitation function are well described by the simple analytical expressions.
Our results are in contrast to [6] where a much bigger effect of the order of 30 to 40
percent was found by comparing different models. It is difficult to assess how much of the
reduction is caused essentially by higher order electromagnetic effects or by differences in
these models. In both cases, the total contribution of higher orders to the cross section is
negative.
Finally, let us make some remarks about post-acceleration. A semiclassical model might
suggest that the parallel momentum distribution of the core is shifted towards larger values
due to an “extra Coulomb push” see, e.g., [9]. However, this turns out to be wrong. In the
sudden approximation, the core-neutron-binding is negligible and also on its way towards
the target, the core alone (and not the bound core-neutron system) feels the Coulomb decel-
eration. Formally, this is easily seen: In the sudden approximation, the momentum transfer
points exactly to the direction perpendicular to the trajectory, the excitation amplitude
depends only on ~q ∆p~ (cf. [8]). This is symmetric with respect to the plane perpendicu-
·
lar to the beam direction. Corrections of this simple result due to small values of ξ were
studied in [8]. They were found to depend only on the phase shift of the neutron s-wave.
This phase shift is given in the analytical model by δ = arctan q. It is a rather delicate
0 − η
8
quantal interference effect and even has the opposite sign to what one would have thought
“intuitively”. Large values of ξ correspond to large values b where the strength parameter
is small. Therefore, higher order effects are not so important. Indeed, in Ref. [14] no effects
of post-acceleration were found for the 11Be system.
IV. CONCLUSIONS
We have studied the basic example for the Coulomb dissociation of a neutron halo nu-
cleus. From the simple zero-range wave function of a loosely bound system it becomes
directly obvious that the low lying E1 strength is an immediate consequence of the halo
structure. It is probably the most beautiful manifestation of the halo nature. Higher order
effects can be described by analytical formulas. This allows a very transparent discussion
of the effects. Our results can be easily applied to all neutron halo Coulomb dissociation
experiments. They are a useful guide for the much more elaborate numerical solutions of
the time-dependent Schr¨odinger equation. Our simple considerations are corroborated by
these more sophisticated approaches. We conclude that higher order electromagnetic effects
are not a significant problem in medium-energy Coulomb dissociation experiments and can
be kept under control.
ACKNOWLEDGMENTS
The authors would like to thank R. Shyam for useful discussions and B. A. Brown and
P. G. Hansen for helpful remarks on the manuscript. We are grateful to T. Nakamura for
providing us with the experimental data. Support for this work was provided from US
National Science Foundation grant numbers PHY-0070911 and PHY-9528844.
9
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